This invention relates to non-linear applications of transistors in uses for computing powers of numbers represented by analog signals and similarly computing roots of analog represented numbers. More particularly this invention shows the use of transistors in non-linear circuits operating in connection with operational amplifiers to compute functions in the form of output analog signals from input analog signals.
Various non-linear response characteristics are useful in machine or computer control systems, usually for the purpose of compensating for existing non-linearities such as an exponential function to linearize a logarithmic characteristic and a square root function to compensate for a square response characteristic. The exponential variation of a semi-conductor diode current with junction voltage has been used. However, surface leakage in semi-conductor diodes and recombination currents detract from the purity of the observed characteristic since there are underlying complexities to the response. The collector current of a diffused base, bipolar transistor is a more nearly exact exponential function of the base-emitter voltage. This is because the complexities, previously mentioned, associated with the diode current and junction voltage relationship, only contribute to the base-emitter current function.
U.S. Pat. No. 3,152,250 issued Oct. 6, 1964 to G.E. Platzer and U.S. Pat. No. 3,423,578 issued Jan. 21, 1969 to G.E. Platzer, et al. show the use of the non-linear response characteristics of transistors for the purpose of controlling industrial equipment. However, these devices do not possess the inherent advantages of the present invention, as will be described further.
As further background to this invention, FIG. 1A shows a common base configuration transistor 10 in a form combined with an operational amplifier 12, shown by the functional triangular symbol commonly employed. This is a standard, known type of circuit for using an operational amplifier to produce a non-linear function based on the characteristics of the transistor. In the situation where a logarithmic function is desired, an input voltage across input resistor 14 produces a linear current response at the output 16. With high gain, the amplifier input voltage must be nearly zero so that an output base-emitter voltage is produced to duplicate the same current in the collector of transistor 10. This voltage, as previously mentioned, is an accurate logarithmic function of the current and thus of the input voltage.
In FIG. 1B, an example of an exponential function is shown in which the input voltage across the base-emitter diode of transistor 18 produces an exponential current which is duplicated in the feedback resistor 20 giving an exponential output voltage at 22. Because of the temperature dependence of base-emitter voltage, the circuits shown in FIGS. 1A and 1B, if used seperately, must be temperature controlled to be useful. The greatest utility occurs when the circuits shown in FIGS. 1A and 1B are combined using matched transistors sharing a common heat sink to allow temperature compensation of the combined circuit. Since the transistor in the one case is NPN and the other is PNP, close thermal coupling as in a common integrated circuit is difficult to achieve.
Referring now to FIG. 2A, a conventional prior art square root circuit is shown. This is because the ratio of resistor 30 to resistor 32 is 1:2 so that a gain of one-half divides the logarithmic voltage generated at transistor 34 to produce an output current and thus a voltage at resistor 32 which is a square root function of the input. Of course, any reasonable power or root function can be produced by varying the resistor ratio, including fractional powers or roots using this circuit. Also additional input logarithmic functions can be added or subtracted with various weighting values to make complex functions possible. The resistance values associated with the summing amplifier 36 must be accurate for precise functional operation of the circuit, but the input and output resistors only affect linear gain.
Referring now to FIG. 2B a prior art square root circuit is shown. This circuit is an example of a circuit in which simple functions such as square root, reciprocal and the product of two voltages are produced without using the extra summing amplifier as shown in FIG. 2A. Transistor 40 generates a reference collector current which is applied to the two base-emitter junctions in a series combination. This combination of base-emitter junctions in series is shown by transistors 42 and 44 as well as transistors 46 and 48, respectively. A fixed current through the lower junction, that is transistors 44 and 48, respectively, of each series fixes the lower junction voltage while the reference current is bypassed by the collector of the upper transistor, that is transistors 42 and 46, respectively.
Any change in reference current produces a change in voltage across the upper junction. This change is divided by the second pair of series connected base-emitter junctions producing a current equivalent to half the logarithmic voltage, in other words a square root current function. The second reference current is converted to a square root voltage function by the second operational amplifier 50. This square root arrangement which is shown by D.T. Smith, of the Clarendon Lab, Oxford, England, in an article entitled "A Square Root Circuit to Linearize Feedback In Temperature Controllers," published in the Journal of Scientific Instruments, 1972, Volume 5, Pages 528-529 shows that equal numbers of PNP and NPN types of transistors can be matched in two separate transistor strings. However, this circuit, unlike the type shown in FIG. 2A having three operational amplifiers, uses base-emitter current as a reference rather than the more precise base-emitter voltage and collector current relationship. In the input signal conversion process, the current which balances the input current includes leakage-recombination current components which cause the output reference current to be smaller than required for the precise mathematical relationship desired. Some of this too small output current is again converted to leakage-recombination components in the output string of amplifiers thus producing a voltage that is also too small. This causes a degradation of results. A similar version of the circuit shown in FIG. 2B subtracts the input logarithm from a reference constant to produce a reciprocal function. To produce a squared result, it is necessary to connect two circuits in series to add logarithms making a product function with common inputs.
While the foregoing has represented a lengthy review of the prior art for this invention, it has been necessary to discuss this in sufficient detail to show the advantages of the present invention. As will be made clear in the following discussion of the present invention, great advantages are obtained by the new type of circuit presented herein. To some extent the advantages of the present invention over the prior art result from subtle complexities of the circuit.